Magnetic Resonance Imaging (MRI) is a medical patient diagnostic technology for noninvasively producing high quality images. In an MRI sequence, a uniform magnetic field, B.sub.0, is applied to an imaged object along the z axis of the Cartesian coordinate system. The effect of the B.sub.0 magnetic field is to align the object's nuclear spins along the z axis. In response to radio-frequency (RF) pulses of the proper frequency oriented within the x-y plane, the nuclei resonate at their Larmor frequencies according to the following equation: EQU .omega.=.gamma.B.sub.0 (1)
where .omega. is the Larmor frequency and y is the gyro-magnetic ratio which is a property of the particular nucleus.
In the well-known slice selective RF pulse sequence, as illustrated in both FIGS. 1A and 1B, a magnetic field gradient along the z axis, Gz or G.sub.slice, is applied at the time of the RF pulse so that only the nuclei in a slice through the object in the particular x-y plane are excited to produce induction signals. After the excitation of the nuclei, magnetic field gradients are applied along the x and y axis, and a magnetic resonance (MR) signal is acquired. The gradient along the x axis, Gx or G.sub.frequency, causes the nuclei to precess at different resonant frequencies depending on their position along the x axis; that is, G.sub.frequency spatially encodes the precessing nuclei by frequency. Similarly, the y axis gradient, Gy or G.sub.phase, is incremented through a series of value and encodes y position into the rate of change in phase as a function of Gy gradient amplitude, a process typically referred to as phase encoding. From this data set, an image may be derived according to well-known reconstruction techniques such as the Fourier transform. It should be noted that it is also possible to obtain three-dimensional volumetric MR images without using slice selection, as known in the art.
When MRI is combined with magnetic resonance spectroscopy, images of different chemical species can be obtained. This procedure is termed Chemical Shift Imaging (CSI) and has great diagnostic values. For example, CSI may be used to obtain separate images of water and fat for each pixel of an imaged object.
Water-fat imaging takes advantage of a slight difference in Larmor frequency between protons in water and fat. Each pixel of a reconstructed image can be modeled with two magnetization vectors representing protons associated with water and fat, respectively. These two vectors are rotating at different angular frequencies with a known difference, or chemical shift, of 3.5 ppm (220Hz at 1.5 T). The two rotating vectors can therefore interfere constructively or destructively, depending on the relative direction or "phase" of the two vectors. The relative phase, or phase shift, of the two vectors can be controlled by adjusting the timing parameters used in an MRI pulse sequence, so that one vector evolves in phase with respect to the other by a certain angle before the image is acquired, as more fully described below. The acquired complex image I, representing the vector field of transverse magnetization in the rotating frame at the time of data acquisition, or at the echo-time (TE), can be written as EQU I=[W+exp(i.alpha.)F]exp(i.PHI.)exp(i.THETA.) (2)
where W and F are real and non-negative variables representing the magnitudes of the magnetization vectors of water and fat respectively, a is the phase shift between the two vectors, .THETA. is an unknown phase error due to B.sub.0 magnetic field inhomogeneity, and .PHI. is another unknown phase error which will be discussed in detail below. Relaxation effects are ignored as they can be absorbed into the variables W and F. Both .alpha. and .THETA. are proportional to a time shift .DELTA.t in a pulse sequence as follows: EQU .alpha.=.DELTA..omega..DELTA.t (3)
where .DELTA..omega. is the difference in Larmor frequencies between water and fat, or approximately 220 Hz in a 1.5 Tesla (T) polarizing magnetic field B.sub.0. (The Larmor frequency of protons in water is approximately 63.9 MHz at 1.5 T, and the Larmor frequency of protons in fat is lower than that by approximately 220 Hz); and EQU .THETA.=.gamma..DELTA.B.sub.0 .DELTA.t (4)
where .DELTA.B.sub.0 is the main magnetic field inhomogeneity. The time shift .DELTA.t in both equations (3) and (4) is measured from the refocusing time t.sub.0 to the echo-time TE, as known in the art. The parameter t.sub.0 is defined differently in a spin-echo pulse sequence and in a gradient-echo pulse sequence, as illustrated in FIGS. 1A and 1B, respectively. In a spin-echo sequence, the refocusing time t.sub.0 is simply the Hahn echo location determined by the 90.degree. and 180.degree. RF pulses; while in a gradient-echo sequence, the refocusing time to must satisfy EQU .DELTA..omega.t.sub.0 =n2.pi. (n=integer) (5)
In other words, in a gradient-echo sequence to is defined as a time instance at which the water and fat vectors are phase shifted by n 2.pi. and, thus, pointing in the same direction. Accordingly, in a spin-echo sequence of FIG. 1A, .DELTA.t can be adjusted by changing the locations of either the 180.degree. RF pulse or the G.sub.frequency structure including both a data acquisition window 2 and a readout gradient 4, or both; while in a gradient-echo sequence, .DELTA.t can be adjusted by simply changing the TE.
The unknown parameter .PHI. in equation (2) is a static phase error which is usually attributed to RF field inhomogeneity or data sampling window off-centering as known in the art. Concomitant field components could also contribute to this phase error. If the image is acquired with a gradient-echo sequence, .PHI. also contains another major phase error .THETA.=.gamma..DELTA.B.sub.0 to due to the main magnetic field inhomogeneity.
In the well known CSI techniques originally proposed by Dixon (hereinafter "Dixon method"), two images are acquired with the water and fat components of the images in phase, and out of phase by .pi. radians, respectively. (See W T Dixon, "Simple Proton Spectroscopic Imaging", Radiology, 153, 189 (1984)). Based on equation (3) and that the differential Larmor frequency .DELTA..omega. between water and fat is known to be 3.5 ppm, time shifts .DELTA.t for accomplishing these phase shifts can be calculated to be zero and 2.25 ms, respectively, at 1.5 T. The two complex images thus obtained, one with water and fat in-phase and the other with water and fat opposed-phase, can then be written as EQU I.sub.1 =(W+F)exp(i.PHI.) (6) EQU I.sub.2 =(W-F)exp(i.PHI.)exp(i.THETA.) (7)
According to the angle between the W and F vectors, the above described data acquisition method can be termed a (0.degree., .pi.) or (0.degree., 180.degree.) sampling scheme. If the phase error .THETA. is determined, the W and F can be subsequently solved from equations (6) and (7) by simple addition and subtraction of the equations. Since both W and F are non-negative and real, one can write EQU 2.THETA.=Arg[(I.sub.2 I.sub.1 *).sup.2 ] (8)
where the operator Arg [ ] returns the phase of the complex number in the square bracket, and "*" represents the complex conjugate. From equation (8), .THETA. can be directly determined provided it is within the range of .vertline..THETA..vertline.&lt;.pi./2. Due to the periodicity of trigonometric functions, if .vertline..THETA..vertline.&lt;.pi./2, the measured phase 2.THETA. of equation (8) "wraps around", as termed in the art and, thus, the actual value .THETA. cannot be uniquely determined. Because the 3.5 ppm chemical shift between water and fat corresponds to a phase shift of .pi. radians, the condition of .vertline..THETA..vertline.&lt;.pi./2 implies that the main magnetic field inhomogeneity must be less than 1.75 ppm over an entire field-of-view (FOV). This is a very challenging condition to meet even with careful shimming of the magnet that is used to produce the magnetic field. Furthermore, even if the original B.sub.0 field was perfectly homogeneous, once the patient is inserted into the field, the magnetic susceptibility variation of tissues can induce significant field variations, which can well exceed 1.75 ppm. Therefore, the Dixon method typically relies on a successful "unwrapping" of the phase map of 2.THETA. to obtain an unique value of phase .THETA.. Phase unmapping methods as currently available, however, are sensitive to noise and artifacts, heavily rely on the spatial continuity of an image and, thus, often become inoperable when tissues in the FOV are imaged as disconnected fragments.
A need exists for a method which can unambiguously resolve separate images of different chemical species based on MR signals. Preferably, such method will not rely on a phase unwrapping method, and will have a high tolerance to noise and artifacts typically encountered in a standard MRI.